thermal effectiveness of a split-flow heat...
TRANSCRIPT
THERMAL EFFECTIVENESS OF A SPLIT-FLOW HEAT EXCHANGER
by
M. IQBAL
Thesis submitted in partial fulfilment of the
requirements for the degree of Master of Engineering.
Department of Mechanical Engineering,
McGill University,
Montreal.
April 1961.
- i -
SUMKARY
A mathematical expression has been derived for the
effectiveness of a split-flow exchanger in terms of C /C min. max.
and NTU Curves have been drawn for f. versus NTU for max. max.
various values of C i /c m n. max.
The effectiveness of a split-flow exchanger has been
compared with that of a reverse flow exchanger and it is observed
that for low values of NTU below 2, there is practically no. max.
difference between the two exchangers.
However for values of NTU greater than about 2 max.
the curves for effectiveness of a split-flow exchanger start
drooping significantly from those of the reverse flow exchanger.
Another interesting result of this study is the fact that the
effectiveness of a split-flow exchanger is improved by having
the heat capacity rate (W Cp) of the shell side fluid smaller
than that of the tube side fluid.
A split-flow exchanger was also built and some points
on the effectiveness curves were checked.
ii
Acknowledgements
The author wishes to express his sincere gratitude to
Professor J.W. Stachiewicz for his guidance and advice.
Thanks are also due to Mr. F. Corrick for his assistance
in the construction of the apparatus.
The financial assistance given by the National Research
Council is gratefully acknowledged.
iii
TABU: OF CONTENTS
Summary
Acknowledgements
Table of Contents
Nomenclature
Introduction
Assumpt ions
Development of an expression for effectiveness of a split-flow
exchanger.
Effectiveness of L.H.S. of exchanger
Effectiveness of R.H.S. of exchanger
Overall effectiveness of the split-flow exchanger
Determination of correction factor Ft
Experimental equipment
Precautions
Test Resulta
Sample Calculation
Discussions and Conclusions
References
Curves of effectiveness of split-flow exchanger
Curves of effectiveness of split-flow exchanger vs. reverse-
flow exchanger
Curves of correction factor Ft for split-flow exchanger
APpendix: Equation (2-31)
Equation (2-3la)
ii
iii
iv
1
13
12.
14
26
44
58
61
69
71
72
74
77
78
79
80
81
82
iv
NOMENCLATURE
Roman Letter Syuabols
A
c
e
q
t
u
w
~otal heat transfer area of the ,tplit .. fl'* ex changer
half of the total beat transfer area of the split-flow exchanger.
flow stream capacity rate (W.~)
specifie beat at constant pressure
baae of natural system of logarithms
temperature correction factor
heat transfer rate
temp-erature
unit overall thermal conductance
- mass flow
Roman Letter Constants
B = ..J!.... 2Ct
D u
= ~ sh
F = tsl - ttl
G = ts3 - t to
H = ts3 - tt3
J = tsl - t t2
·~ 2
ft
ft2
BTU/hr oF
BTU/lb oF
RTU/hr.
lbs/br.
Kl' ~, K3
, K4
, KS' K6' "! and K8 = constants in differentiai equations.
L = -V e
=
v
M 0 "/ -Y = (e - e )
N = (1 - p) e"~ + (1 + p) e -Y
n = NTUmax
tt3 - tto p =
~s:l. - tto
tsi - t ct R = sg. =
tt3 - tto cs
s = 2 eDAh
t ts2 + ts3
= 2 sq
-t.:rr.Umax}l
2
x = 1 + +(cmi~)
e cmax
1 emin
x = +--cmax
-ImJmaxjl 2
+(Cmin) y = 1 - e Cmax
~in y = 1 - Cmax
Greek Letter Syœbols
E 2sh
é.s
d•notes difference
exchanger effectiveness dimensionless
effectiveness of L.H.S. of split-flow exchanger
.effectiveness of R.H.S. of split-flow exchanger
effectiveness of R.H.S. of split-flow exchanger when ct = emin.
effectiveness of R.H.S. of split-flow exchanger when csh = emin•
effectiveness of complete exchanger based on él and é2 when ct = emin•
effectiveness of complete exchanger based on é. 1 and ê.. 2 when C8 = emin.
vi
Greek Letter Constants
o( B + D = Js2 + o2
B -t D = J s2 + o2
= AhjB2 + D2
? B = Ja2 + o2
t D =
J + D2 s2
" = D ~ _ft2 + o2
Dimensionless Groups
= c.,. Flow stream capacity rate ratio
~ heat transfer effectiveness of an exchanger; a function
of NTU, CminlCmax' and flow arrangement.
NTUmax - n&~ber of heat transfer units of an exchanger, a heat
transfer parameter. (UAh \ or (JlL_) Cmi~} ~in
Subscripts
Ah hal f are a,
a left end of exchanger
b middle of exchanger
c counterflow
f r ight end of exchanger
p parallel flow
vii
r ratio
s shell
sb shell half
t tube
t .c tube .. counterflow
tp tube - parallel flow
tr tube .. end where flow reverses .
w weighted
0 1 2 3
for temperature subscript notation refer to each
figure individually
- 1 -
INTRODUCTION
In a heat exchanger the rate of heat flow from the hot
to the cold fluid is proportional to the temperature difference
between the two. For design purposes it is essential to know the
mean difference in temperature between the inlet and exit tempera-
tures.
The Log-Mean Temperature Difference Aperoach.
The classical approach to the performance of a beat
exchanger is in terms of log-mean temperature difference (LMTD)
and a non-dimensional correction factor Ft• Expressed in the
usual nomenclature, the rate of heat transfer for unit time is
given by
q = U.A.Ft. LMTD (counter-flow).
If the exchanger is actually a counter-flow unit the factor Ft
is unity. For all other flow arrangements Ft is less than unity.
In tubular exchangers the simplest case of counter-flow
or parallel-flow is that of two concentric pipes(Figure 1 and
Figure ~· The LMTD for these two cases is easily derived and is
given by:
- 2 -
GOgN!ER-FLOW EXCHANGER
c
ct~ ---Ct
lj--:! 1 ===~-======r-I Cs
SYSTEM DIAGRAM AND
EXPECTED TEMPERATURE DISTRIBUTION
FIGURE - 1
- 3 -
PARALLEL-FLOW EXCHANGER
c
~-----~-----'1 (1 --
SYSTEM DIAGRAM AND
EXPECTED TEMPERATURE DISTRIBUTION
FIGURE - 2
LMTD (counter-flow)
LMTD (parallel-flow)
- 4 -
= (tsl- h·3) -( fj4- tf:o) iM. Es,- tt"I.
ts4 - é to
= (ti~-tto) -(Cs4-tt:?>) 1.,..,_ t ~ 1 - t to
ts.4- tB For any set of terminal temperatures, the LMTD for parallel-flow is always
less than that for counter-flow unless the temperature of one flu!Ld stream
is constant throughout the exchanger.
In practice, in the majority of industrial installations, the
counter-flow beat exchanger is not as economical as multi-pass or cross-flow
units. In multi-pass exchangers the flow is partly counter-flow and partly
parallel-flow and as a result the LMTD lies somewhere between LMTD (counter~
flow) and LMTD (parallel-flow). The same is true for cross-flow excb.a.n.gers.
For multi-pass exchangers. Nagle (1) derived in detail the equations
for mean temperature difference (MID) for the 1-2 exchanger, i.e. having one
pass shell-side and two passes tube-side. Figure - 3. He also extended the
derivations for the 2-4 exchanger, i.e. having two passes shell-side and four
passes tube-aide. Graphical and trial and error solutions were obtained for
these equations and curves of Ft versus dimensionless temperature ratios
R t~, - t!:.4
- tt!> - E ~o tt"!> - t to
t~, "'"""~l:o
were plotted.
* Underwood (2) solved Nagle's equations for MTD. Later Bowman (3)
extended Nagle and Underwood's work by developing equations for any number
of shell passes and plotted curves for Ft for 3-6, 4-8 and 6-12 exchangers.
* This paper was not available, however, the information is based on
reference (7) .
- 5 -
1 - 2 EXCHANGER
(Reverse-Flow Exchanger)
~----------------------------~-:_.__ ct ~-------------------------------~ Ct
SYSTEM DIAGRAM AND
EXPECTED TEMPERATURE DISTRIBUTION
FIGURE - 3
- 6 -
In cross-flow exchangers, the fluids flow at right angles to each
other, therefore, neither the counter-flow nor the parallel~flow equations
for LMrD can be applied. Mathematical analysis of the MTD in single pass
* cross-flow and double pass cross-flow were given by Nusselt (4) and Smith (6).
Bowman-Mueller-Nagle (7) co-ordinated the results of all the above
mentioned types of shell-and-tube exchangers and cross-flow exchangers, and
presented in a single paper the correction factor curves for various surface
and flow arrangements.
The NTU Aperoach
In the thermal analysis of the various types of beat exchangers
presented above, the equation~= UA Ft LMTD (counter-flow) is used and is
found convenient when all the terminal temperatures necessary for the
evaluation of the appropriate mean temperature are known. There are however
numerous occasions when the beat transfer area A and the overall beat trans-
fer coefficient U are known or the latter can at least be estimated, but the
temperatures of the fluids leaving the exchanger are not known. This type
of problem is encountered in the selection of a beat exchanger or when the
unit bas been tested at one flow rate but the serviae conditions require
different flow rates for one or both fluids. The outlet temperatures and
the rate of beat flow can only be found by a rather tedious trial and error
procedure if the charts of Ft are used. In such cases it is desirable to
circumvent entirely any reference to the logarithmic or any other mean
temperature difference.
* This paper was not available, however, the information is based on
reference (7).
- 7 -
This method bas been proposed by Nuaselt (5)* and later by Ten
UA Broeck ( 9-) who showed that there is a relation involving Ct' R, P and Ft
in the equations
also
q = UA Ft LMTD (counter-flow)
= UA. Ft . (ts1- b:-s )-lts4- t~) l tsl - t~
t$4 - tt:o
therefore from the above three equations one can Write
UA _ k( ';:.R:) Ct ft l \- R. )
which connecta the four dimensionless parameters, and is independent of any
considerations of flow arrangement.
From Underwood's equation of Ft for one shell pass and two tube
passes,
Ten Broeck obtained
~t = jRl-+1 ~Ll1-P)/(1- RP)J
(R _1 )l ?.-P(R+1-)R'+1 ) 2- Pl Rt1 +)R';-1 )
l_ 2 -P ( R-t 1-JR~+1 ) 2 - P l R + 1 +) Rl.-+ 1 )
connecting the three dimensionless parameters, and plotted the curves for
UA P versus Ct with R as the third parameter. P the dimensionless temperature
UA ratio was called a thermal efficiency and Ct bas been subsequently termed
as NTU, the number of beat transfer units. Ten Broeck also plotted similar
curves for 2-4 exchangers.
* This paper was not available, however, the information is based on
reference (8}
- 8 -
Defining again the thermal effectiveness as
~ _ actual energy transferred ~ - maximum possible energy transferred
_ Ct ( b:"3 ~ b:o) C& ( ts.,- t ~4) - C~~~~ l ts, - t to) CW\i'r\ l t~, - t to)
and NTU max
UA c min
c and third parameter min, c max
for a large number of flow arr~ngements 1 Kays and London (10) have given the
curves and in some cases also the mathematical expressions for effectiveness
versus N'l'Umax in terms of the third dimensionless parameter ~in. ~ax
For some of the more common flow arrangements the expressions for
effectiveness are reproduced below:
Counter-flow - HTUh\LtX ( 1 -
E-=- 1-e ( . -NTU~ ( 1-
1-~e. c~
Parallel-flow
-NTUmo.x. ( 1 +
E 1- e. ---------~-------1 -t c "WW~
Parallel-Counter Flow c.~
One shell pass, two tube passes
-NTV~:~}+ (~~~ \2. 1- e. ~J
- 9 -
Cross-Flow
for Cmin = Cmixed
and ~ = Cunmixed
for Cmax = Cmixed
and Cmin = Cunmixed
E. = Cw.ox c~w.
Split-Flow Exchanger
Split-flow exchangers are often used when the permisaible pressure
drop of the shell-side fluid is so small that the fluid cannot be permitted
to travel the full length of the shell. The fluid is admitted at the centre
of the shell and the flow is split as shown in Figure 4.
The pressure drop through the shell of an exchanger may be approxi-
mately considered as directly proportional to the length of the path and to
the square of the maas velocity of the shell aide fluid.
Sinca in a split-flow exchanger, both the maas valocity and the
length of the path are reduced by a factor of 2, the resultant presaure drop
is approximately one eighth that of a conventional exchanger.
- 10 -
SPLIT-FLOW EXCHANGER
ts2
ts3
tt21 tt3
t tr LI
SYSTEM DIAGRAM AND .
EXPECTED TEMPERAXURE DISTRIBUTION
FIGURE - 4
- 11 -
Kern and Carpenter (ll) darived an expression for the mean
temperature difference in a split-flow exchanger. This is an implicit.·
and tedious expression and lends itself to solution by a trial and error
method only. In practice thus, an engineer seldom can make use of it
and generally employa the curves of l-2 exchangers.
Schindler and Bates (12) have drawn Ten Broeck charts for a
1-2 divided-flow beat exchanger, however, a study of the effectiveness
versus NTUmax relation and correction factor of this relatively impor
tant split-flow exchanger bas not appeared in the literature heretofore.
- ll -
Development of an Expression for the Effectiveness of a
Split-Flow Exchanger as a Function of Capacity
Rate Ratio and Number of Heat Transfer Units
The system diagram and expected temperature distribution for a
split-flow exchanger is shown in Figure - 4.
The split-flow exchanger is es•entially a combination of two
independent counter-flow parallel-flow exchapgers: · For thé mathematical
derivation, the two halves, (L.H.S. and R.H.S.), are considered separately
and then a combined effectiveness is derived.
\
- 13 -
ASSUMPTIONS
The following assumptions are made in development of an expression
for the effectiveness of a split-flow exchanger.
1. Equal weight flow of shell fluid in right band side and left band aide
of the exchanger.
2. Constant specifie beats for both fluids on both the left and the right
band side of the exchanger.
3. Equal heat transfer area in each pass and on both aides of the exchanger.
4. Constant and equal overall coefficient of heat transfer U on both aides
of the exchanger.
5. Shell fluid completely mixed at any cross section.
6. No phase changes within the exchanger.
7. Negligible heat losses to the surroundings.
8. The excha~ger operating at steady conditions.
- 14 -
1 - Effectiveness of L.H.S. of Exchanger
This derivation is based mainly on an unpublished report of
c.e. Wright (13) excepting that the shell-side fluid flows from right
to left.
Consider the energy balance and rate equations over the area
dA, Figure - 5.
Energy balance:
Rate equations:
dq,. = dqs = dqc. + dq,.
d\ = C.s.J-. Jts
dq~ = Ct dJ. tc. = U ~A (A tc.)
dq~ = -Ct d.t tp = Uf' (6. t ~)
(1-1)
(1-la)
(l-1b)
(1-lc)
Substituting the rate equations (1-1b) and (1-1c) in (1-l)
d.~A' =Ct lJttc. - ~t tp)
since
and from Figure - ~
he nee
or
also
= UiA ( b tp + ôte.) ..... (l-l)
clt tc. - ~t tp = J.l b:c_ - t '=~J t tc. - tt~ : At p - A tc
c~ J (~tr -A~c.) = u~ lAtp + Ab:) dJAt~ - 6tc.) _ JL ( Atp + b.h:_)
cl.A - 2Ct \.
C~ J.ts = Ct ( J..ttc. -dJt~)
d.ts • ~ ( dtt( -Jtt ~) ..... (1-4)
- 15 -
L.H.S. OF SPLIT-FLOW EXCHANGER
t----- A ---- --t--
TEMPERATURE DISTRIBUIION
FIGURE - 5
- 16 -
Subtracting dttc from both sides of (1-4)
dis -Jttt :: ~ clltt- Ss_ cll.k:.p - Jk tc. (1-5) '-~ c.~
aince dJ:~ - d.t tC. :. J l t~- ltc. J ::. J.~l:c..) and from (l-1b) d.t~ = Ud.A A ~c.
2Ct
from (1-1c) d.( ~~ ~ - UJ.A Â 1:. b 2,Ct r
Subatituting the above three equations in (1-5)
Now substracting dtt from both sides of (1-4) p
J.b,,- J.l:tt e ~~ Jtt-c - t_ dl:t~ _dhp • • • •• (1~7)
Since JJ:s - dtt~ = cllb~ -t~p) = cl(_A~~) and from (1-1 q,) Jt tC. = UJ..A. ~tc.
2G-
from (.1-1 c) c{h~ =- ~~ Al:~
- 17 -
bence substituting these three equations in (1~7)
u u Subatituting B = 2Ct and D = 2Csh in (1-3), (1:'"6) and (ltr-8)
J. ~~~) - J.~;-) = l'> (_Al:~) + ~ (t>A:c) ••••• (1-3)
••••• (1-6)
• • • • • (1-8)
àifferentiating the above equation
d_ ( .ô.t:-~) __ ( D-t J cÀ. ( t:kc.) \ dt(~ dA - ]) 7 dA + J)ll. d..A"L
Subatituting the a'bova two equations in (1•3) and •imp1~fying
or 00000 (1-9)
.... 18-
Similarly from •quation (1-8)
Equations (1~) ~nd (1•10) are differentia1 equations whose solution is
( 1\ ~~
.À~l.DÀ-B J K l. =o
where
( lH·,.{B.':l.+~ )A (f) -.Jal--+'D ... )A A~<. :. k., e.. -T K:t e..
••••• (1-11)
(~ +)&l.-t'Ji1- )~ (D -J!i>-+~1)A A~ p = K ~ e. + Ktt t. ••••• (1-12)
The constants ~ 1 K2, IS and K4 may be determined from the boundary conditions.
From Figure -+ .S: when A = 0 1 Ab:. = A~p = f~l. _ t t'f ••••• (1-13,14)
-
- 19 -
Also from ~quation (1-6)
••••• (1-15)
Also when A = o, equation (1.-U) gives
~t =K +K-=t ... t c 1 -~ s2 tr
••••• (1-16)
Note:· In the foregoing and subsequent equations in th.is article, A does
.»Dt .represent the total area of the split-flow exchanger, but representa
bhe variable area A, which can assume any value 0 - ~·
Equatiug equations (1•15) and (1-16) 1
(2!>-S)(h.l.-rt--t) = ~,(D+)B'"Jrll"l. )~~l.(~.JB,.~l>-a.)
-:::. D ( \.<' -t K~) -t l:f::.' - 'f:-"1.)} 'è;}· -t r:?
= J) (t~2.-~tt) +(t,-'t(~)Js~ 1?
or ~-B) (hl.-~~1) =J P.}·+ ri" ( \<\ - K0 \<, -\<1. = .D-B (h~1. -tt-'~-)
J B"' ;- J.)"l-••••• (1-17)
- 20 -
• • • • • (1-18)
••••• (1-19)
Wow K3 and K4 must be eva1uated.
a1so equation (1-8) at this boundary condition becomes
: (?.J> 1-B)( ts~- ~ tYJ oH o o (1-20)
A:o
Differentiating (1-12), putting A= 0 and equating it to (1-20) in a simi1ar
way as before, it can be shown that,
(zl -+S)( t~1 - trt) == K~ (:D +)B~Jf-) + \(lf \_}-)B,_-+~1.)
or
as
. ..
(D +B )( ts2.- t~t) = jB~+y. (~1.- KttJ
K ~ - KLJ c l> + ~ 1 c ~2- tt -t \ j~} -t-~2. \ J
K'l+ ~ : tsz. - ttt K = t~- b~Y \ 1 + 1) + E> 1
~ ~ l j·p/·.JrJ>4 j
\( = ts>.- !:tv ( 1 _ b-I-t'> (
~ 2 l J ~l.~~ j
• • • • • (1-21)
• • • • • (1-22)
Substituting the values of the constants K11 E11 K3
and K4 in (1•11) and (1•12)
(, ~ ( V-Y, ~ ]),4. \ J !>~-+D1 A -)B .. ~l)" " }
e. \\+ ~ e_ ~ ' - J~}··H):a.) f(_
- 21 -
the ab,.-ve. are general equations where A is at ill a variable.
The a'bo.ve temperature difd!al!'enee ·equations for the centre of the exchanger
i.e. at b when A = ~~ill be
B+D Subatituting o( = -;=::==
)B4 -+b'il..
..... (1-23)
• • • • • (1-24).
The next step is to write expressions for effectiveuess in term.
of capacity rate ratios and temperature difference and then to link the last
two equations.in order to find effectivenass in terms of dimensionless
parameters.
- 22 -
f 1 =The effectiveness of L.H.S. of the split-flow exchanger
= Actual beat transfer rate. Maximum possible beat transfer rate
Maximum possible beat transfer rate = ~min(t 81 - ttl)
:. E 1 ~ Ct (tu. - t ~~ )
C .... w.(ts,- ~t-r)
"" L E., c"""~ L L L.S~ - J...\'\ Cû.. l!>,- Ltl = --------- - b:1
: · ts, - t t 1 :.
1- é1 c""~ Cu.
a1so from (1•25) tt2. =tt\ t (ts, _tt,) E, C~ Ct
Subtracting (1•28) from (1-26) and using (1-27)
- ••••• (1-25)
••••• (1-26)
• • • • • (1 ... 27)
• • • • • (1-28)
- 23 -
l:st-l:tl.- ( h,_ h,) - (t .. - h~ f, C~
" ( h,. - ~h) ( \ - é:, ('t) . ~- b-2.
1 C"Mk...
• • Cs\ - b-t = - é.' Ct
From equations (l-23) and (1-24) and Figure - 5
Equating (1-29) and (1-30)
1~ ) y -.Y j _ E C \-\.\iM _ ~ - p e. + ( l + ~) e.. 1 1 4- - t -'1
~ +<t:)e_ + (t-oC )e. . y ~
:. E. c\M~ - 1- (1-~)e. + (\+ ~)<L ·l ct - "" y ù +~)e. + (\ - oC)~
:. [, =
y -Y. (-y -~ o<::(e. -e )+~ e.- e.)
( y .. '{) ("" -'1 e. +~ +oé' e.-e.)
:: ----:----~~-
~ + [ l + 'ë_2.i J -2."(
1- e.
q( + ~
[
-"LY I 1-t- e. -1y
1- e.
•.••• (1-29)
••••• (1-30)
- 24 -
B +D Cw.w.. -;::::::== '1(.. --J &' + t?· Ct-
:--------~~----------~~~ -UAtjq -+~
- +- ""'"" + ~ 1(-1{ --\-~ (u U)C· (~ Up;;:fl 24 2.~ 4 Ct- 2 Ct2. CsL.
2
When C = C ~ C = C , then t min sh max
é,-= ----.---::a.~---N~--J-;::\=+=(=~::=.=)~:::a.'
1+(~~) IH
2
1+ (~MW, + c~
-Hru,_J '+ ( ~~ ' -e.
\+ e.
••••• (1-31)
- 25 -
and Ct = Cmax' the resu1t (1-31) remains unchanged.
emin The curves of effectiveness with the parameters ~
max
and NTUmax have been drawn and are avai1ab1e in referenc~ (1~).
- 26 -
II Effectiveness of R.H.S. of Exchanger
This section is a further development of Section I and is based
on similar treatment as before.
Considering the temperature distribution diagram Figure - 6 and
establishing the energy balance and rate equations in this case in a similar
manner as in Chapter I.
dqA. = dqs = dqc + dq~
dq~ = -c hdt s s
d<lc = -Ctdttc = U~A(.otc)
dq p = Ctdttp = U~A(Atp)
Substituting (2-lb) and (2-lc) into (2-1)
dqA = Ct (dttp - dt tc) = U~A(Atp
since dt - dtt = d(tt - ~ ) tp c p tc
+At ) c
and from Figure - 6 b:p - t I:C. :. A tc - At~ bence Ct J. Cô.tc-At~)=~ (ôf:c+At p)
(2-1)
(2-la)
(2-lb)
(2-lc)
..... (2-3)
Now again substituting (2-la), (2-lb) and (2-lc) into (2-1),
-Cu JJs :-Ct Jlt<. + 4 vltt~
or JJ~ ~ ct CJJ:tc. -J.b:~)
Subtracting dttc from both sides of (2-4)
clt~ -dltc. = g_ J.t~:c.- .ft cfrtt> -cll:tc. Cs~.._ Cu.., 1
since dis -dl tc c:. ol (Cs: - ttc.)
••••• (2-4)
••••• (2 .. 5)
- 27 -
R.H.S. OF SPLIT•FLOW EXCHANGER
TEMPERATURE DISTRIBUTION
FIGURE - 6
- 28 -
and from Figure ... 61 t - tt =At s c c
also from equations (2-lb) and (2-lc)
hence substituting the above in (2-5), in a similar manner as (1-6)
Now substracting dt from both sides of equation (2-4) tp
0 • • • 0 { 2co6)
dls- cLtt:r = _s_ oLt tc. - .fL dtt~ _ dl tp Cst. Cu.
since Jls -clttr ~ J. (ts- C~~) and from Figure - 6 ts.- t tf :. At f
also from equation (2-lb) and (2-lc) JJ:: tc. c:- UJ..A Al:c. 2.Ct
and cll tf = ~~ t::.tr
hence substituting these values in (2-7) and simplifying .in a stmilar way
as (1-8)
• • • • • ( 2-8)
u u Substituting B • 2C and D = ~ in (2-3), (.Z-6) and (2-8).,
t .sh
J. (A tc.) J( .6t~) aA - dA ~
(2-3)
• • • • • ( 2-6)
• • • • • ( 2-8)
- 29 -
from (2·6) h.l p = (B;!> ) A~c - t. cA. ~c)
' d_~A(:A p) = ( B_:l)) rL~~) _ ~ ~ ~~~~c) differentiating this ;_: \. JJ ~ ~... ~-
Substituting these two expressions in (2~3) and simplifying
• • . . . (2•9)
In a similar manner (2•8) and (2-3) yield
• • • • • (2~10)
The solution of the two differentiai equations (2·9) and 2·10) is
(-1> +Jf/·~l)l. )A ( -))-jS~+~ )A ~tc.= ks e_ + ~6 (L • • • • • (2-11)
(-l> +J B·~+ff )A (:-J> -J'6 l. +~~ ) lA
6tr = 1<1 e.. + \:::e ~ • • • • • (2-12)
wbare A as in section I is a.ny area 0 to ~·
The constants Ks' K6, ~ and K8 may be determined from tbe
boundary conditions.
When A= 0; from F~gure- 6 and equations (2·11), (2-12) and
(2-6) the fo11owing re1ationships are obtained:
Atc = ts., -tt-\ - K~ + Kf.
At p = tt1 - tt2. • K7 -t K~
(2-13)
(2.;.14)
(2 .. 15)
.. 30 -
Now differentiating (2-11) and eva1uating for A = o,
= ks t:D+jB'~l>' )+ K, ~J):iB~J>'-) ...... (2-16)
A=o
~quating (2-15) and (2-16)
lB-:b )l t~,- rt-t) -1> ( t-s;,_ h-2..) =~s; +~) (:-1>)+6< ~- ~)J"{f.;i
OJ." 1) ( t~\ - h0 ... 1> ( ~~ - b-2.) -= ( ~s - ~)j B 1. -T :rl
• . . Ks -'\(~ = I!>(h:.,- tt:1) -]) ( h,-h-2..) ) B ,_+.I>l
..... (2-17)
... Ks = (t~, - h-v(Jf01..rJI t~) -(t-~·-h-2.)D ~ B'2..·Jd>"l.
(2-18)
k6 - (t~, - h-JU B'+!l' - 'B)- (t~\ -h·2.Jl> ...•. (2~19) - '21-J f3 1:-T l>'
By a simil•at: proeess the other two constants are obtained:
K;z=(~, - h2.)0>S?- - tt,)-:)(t-s.,-th) ..... (2·21)
2 Y-. J 'B"L.-t !)2..
Ka _lh,.-fn)(j B"l.~D:l. +~)+t>(ht-tb) ..... (2• 22)
2. x.} ~2..-+J>:t. Subatituting these v~lues of constants i n (2-11) and (2• 12),
l-J) -+)'B~r>~- )A
- 31 -
su:bst it ut ing
(2-23)
Similarly it can be shown that
• • • • • (2-24)
The above are general expressions of temperature difference where A is
any area () to ~·
These temperature differences at the extrema right hand ~d at
point f where A = ~ can be obtained by substituting
in (2-33)and (2-24) , so that
-+ e. J ] -i1 t
. ,. .. ·:·
- 32 ...
• • • • • (2-Zla)
Similarly
-l>A.t...)l ~ --1 \ f: ~ 1 (Atr \ = ~ e. l 1-t-t +Je. -Ill+ l'+ :r o~i"Y j
from Figure - 6 ~t~).y -= ts~- t~; -: ~ J
In a similar way as (2-23a), it can be shown that
Substituting
)
the equations (2-23a) and (2-24a) can now be written in a more compact form,
L M G=-F .. -J s s
N M H=-J,.-F s s
. ••••• . (2.-2Ja•i)
• • • • • (2-24a-i)
.. 33 ..
As in the last chapter, the next step is to write expressions
for ef'fectiveness in terms of capacity rate ratios and temperature differences
and then to link the last two equations in order to find the effectiveness
in terma of dimensionless parameters.
62 =The effectivenesa of R.H.S. of the split .. flow exchanger.
Actual heat transfer rate • Maximum possible beat transfer rate
Actual beat transfer rate
= csh(tsl- ts3) = ct(<ttl _ .. tto) + (tt3- tt2))
Maximum possible heat transfer rate = C i (t 1 - tt ) mn s o
- Ct l (tt\- tt-~t(t~-~t-2 ~ - (~ ( t~,- ~~) .....
From (2•25) the following relationship is obtained:
(2-25)
••••• (2-26)
• • • • • (2•26a)
Subtracting (2-26a) from (2-26)~
1:,3-t~?> : t"- hl.+ btl- tro- (t-~,-h.) (s.?~ -tf .. ~)
= (t.,-tto)ll-é,_ ~~ -E,_ ~~""1-(~-lt.) ..... <2-27l
.. 34-
Subtracting t from both sidès of (2-26), to
l,~ - bo = ( ~" - t •o) \1 - f.. t:) Dividing (2-27) by (2-27a),
••••• (2-27a)
( \ -t:2
Cw..~ _ éz. C~;..)- tu- tt-\ \ (~ c~ t~, - ho - • • • • • (2-28)
\-El.. c~ Cs.t.,
lt can b.e written, tt2 - ttl = (t81 - ttl) .. (t 81 - tt 2) = F - J ••••• (2-28a)
and (2g27a) can also be written as
ts~- k .. o G
1-E.. ~ = • • • • • (2-27b)
Subst1tut1ng the values of (2-28a) and (2-27b) in right hand side of (2·28),
(2-29)
from (2·24a-1) and (2·23a-1)
N M S J .. S F H ts3 .. tt3 L M =c;=t -t i F - S J s3 ·· to
• • • • • (2-29a)
Equating (2·29) and (2-29a)
• • • • • (2-29b)
- 35 -
Cross multiplying and rearranging (2-29b),
(1-é. ~-:- E, ~-:-il~ F- ~ J)- (F-J) ( 1-E. t:-) = ( 1-t, ~K ~ L ~ ~
or -E.l ~:+Ct)( 1;~=- ~ ~+E,~-:r)t' +~ ~~(~J-~i)
t ( N-S tM)- (L-~+M') ------------------------------------------
.!~I'J ~ M (c~ Cw..:..) lf. ~ ~j c~ (~ ~~ F 11-+ -G +~ + --\ S- -M--l-~-(~ ~ ~...+- J . C~ (~ 4t..4
• • • • • (2-30)
J Now values of F' L, M, N, S are to be converted in terms of
capacities, capacity rate ratios and NTU and then are to be substituted max .
in (2-30), y -'{
. J ~ - tl-'2.. U -~) e + ( \ -+ ~) ~ from ( 1-30) 1 F = = ---=------{-:---------_-::-:yr
Cs, - t ~ (!-+ o() l. + ( \ _ ~) e.
- 36 -
-2Y \+ .e.. ~ -
-2Y L_ ,_ e. F -If
\ + ~ +cf:.. -tv \ - e..
where
1 + 9:._ - c.SL...
~ J \+ (~J
or
Similarly, 13 =
- 1- ~ J l+(~y
eSt.. - i or ~
c:. ~::=;::::====-
}(~} +1
• • • •• (2-30a)
or
u~ = c
t
- 37 -
.----2.
1 + (~)
_u~JI+ (~Î 1- ~ l+e. -
-~}1+ (~J j 1 +(tl ..... (2-30b) • J = ___!_:' -~e.:_--;J===::;:/".=::r;:---~-
• • F _ UAe.. 1 + { ~ \ + G-lt \ C.st.. C.st..
or =
1 + e.. +
-'*'Jh (:iJ J 1 + ( ~J 1 - e..
-~J l+l~t ~- \ 1
+~~.JI+(~)' - j \+ ~l .... , (2-30c) 1 e..
-~~r ~ -t\ c;j'~\~1 c~ 1 + ~ + )----=~, 1 -~~}+ (~]'" 1+ (~)
- 38 -
Now expanding the other values:
'V -Y y -Y "V -'v) L = (1 t f') ~ -t ( \- f') e. : ( e -+ <2. ) -rf ( t - e
- 39 -
N = 't -Y
(1 -l') <L + ( 1 -+ ,P) ~
.:DAt. s = 2 e.
[])A~}~: ~1 ] j ~~ -t\ = 2 e_
•
or =
}f;-tJ 2e.
~c; +G~ =2e..
J Substituting the above values of F' L, M, N, S in terms of
ct ch c: ~, u~ and cts' u~ in (2-30), equations are obtained for62t and c2sh"
sh
These equations (2-31) and (2-Jla) are placed in the appendix (see pages
81 and 82). as .t~ey :are too lengthy to include in the text.
- 40 -
These equations can be simplified somewhat by dividing them by
(eV - ëY) throughout in numerator and in denominator and then putting
ct emin csh emin --- =---- and --- =---- respectively it is finally obtained, c c c c ,
sh max t max
where n = NTU max
t. J 1+ (t;J c
C _ min
r - C max
J f~c .. ~ ... Y -NTUw.._ f-t- \ ~J
x - 1 + e_
Ct C i mn Equation (2-32) applies only when--- = ~. csli cmax
J
• • • • • ( 2-32)
x..-'
c~ + -c~
- 41 -
csh emin Similarly when C = -C-, the second equation gives:
t max
y--t- 'i +-r- y-+-r- ~ + T 1 + t ~ c ... 21?..\ll-t) '1 t'{.. c, 2 ~'~-~) 1 l ~ + t
E2~~------~----------------------~~--------------" ~'\--t\ ~+a y T
~+x '1 t
1.~ _ ~~ + ; + _Y_-t_T_)L __ 1 ?.. ~ ., \ J t x + ~ "1
'1 1 • • • • • (2-33)
For the special case of isothermal shell-side fluid i.e. for
c ct ~in = 0 = ~~ equation (2-32) reduces to max .sb
:NTU~) c.. ••••• (2-34)
while (1-31) for same condition gives
(2-35)
The temperature diagram for the complete exchanger for such a
case will be as in Figure- 7.
Taking the second special case of isothermal tube-side fluid
i.e. Cm.n Cab
for r. _ = 0 = c:-' equation (1-31) gives the same result as (2~33) -max t
-HTU~
i.e. E. 1 == t.~SJ,.._ = 1 - e..
The temperature distribution diagram for this case for the
complete exchanger will be as in Figure - 8.
- 42 -
SPLIT-FLOW EXCHANGER
t tsl t 82~--------~--------~------------~------, s3
----r--jt3
TEMPERATURE DISTRlBUTION
FOR
ISOTHERMAL SHELL-SIDE FLUID, Ct = 0 cs
FIGURE - 7
to
- 43 -
SPLIT-FLOW EXCHANGER
s3
ttrr-------------------~------------------~tt3 tto
TEMPERATURE DISTRIBUTION
FOR c
ISOTHERMAL TIJBE·SIDE PLUID, ·~. = 0 ct
FIGURE - 8
- 44-
III - Overall Effectiveness of the Split-Flow Exchanger
Refer to Figure - 4 for the analysis in this section.
There are fiv.e distinct possibilities of capacity rate ratios in
complete exchanger.
Possibility 1:
Cs > ct
csh / ct
a) ~total based on shell-side flow,
C, ( ~.~- t,.,. i (o,) t.t =
G: (Es, - tto)
;:::.
=
:: é2.t + [, ts..- tt., E~, .... t-,.o
<4.. ( ts., - tr.l.) 4 (~,-ho)
b) t total based on tube-side flow,
tt = 4(t~- tt-o) Ct (~~-ho}
• • • • • (3-1)
• • • • • (3-2)
• • • • • ( 3-3)
- 45 -
&_ _ Ct_ (t., -lt>)+(h,-l:.to)+ ( t~1- tt,)} Ct tb,- ttc> J
c c t\1 - t~, '=-l.'lt. +c.,
t!.~ - b·o (as before) .•••• (3-4)
Possibility 2:
Csh- Ct
a) <: total based on shell-side flow,
c. ( t,., - t.,. ~ 1-.3 ) Et =
Ct ltSl- t~o) ..... (3-5)
=
4Ll l:s,- ts1) Cg__ l ts, - ts1) = Ct l ~~ - b:o) + C.t-lrs.,- h-o)
..... (3-6)
b) E total based on tube-side flow,
..... (3··1)
(as before) (3-8)
- 46 -
the above equations (3-2), (3-4) and (3-8) apply for any value of
or ct c = 0- .5
8
Possibility 3
a) E total based on shell-side flow,
. ~ l ts, - h.,; Cs.1)
C~ lts,- t~) z:s =
b) t total based on tub~-side flow,
Es = Ct:(tt~- Cro) (~ (h, - b.fl)
• • • • • (3-9)
• • • • • (3-10)
• • • • • (3-11)
• • • • • (3-12)
- 47 -
The expressions (3-10) and (3-12) will apply in the limits
or
Possibility 4:
c = c s t
a) [total based on shell-side flow with Cs
Cs (t .. - b-;_ t.~} ~ ( t~, - tt:c)
~ 4-( t .. - t.,_~(.,,) - 2 est. Cr~, _th)
. 1 l4J.. ( t:., - h.1.) ~ ( k~, - l:Sl.) 1 = 2 est.( tr.,- h·o) + Cse.lt~, -h(lj
in denominator,
01001 (3-13)
• 0 0 0 0 (3-14)
- 48 -
b) é total based on tube-s ide flow with C8 in denominater,
==
ct( h-~ - h-o) Cs ( h.t- ho J ••••• (3-15)
••••• (3-16)
Now redoing the above by taking Ct in the denominator.
a) 6 total based on shell-side flow, with ct in denominator,
c.sl~<l- ts,~ t .. ) Ct (t~\- b-o)
( ( L _ t!>,±t,ll
2 ~\\.SI 2 Ï 4l~~\ -h-o)
Cg__ l ts., - Cs.?> ) ~ l t~, - h . .,_) 4 lh.,- ho) + 4(1:-..,,- b-o)
••••• (3-17)
• • • • • (3-18)
The expressions on the R.H.S. of (3-18) cannot be formed into
effectiveness since Ct is not Cmfn for the individual exchangers. However,
Since 2C8 h is equal to Ct, substituting it in the denominatàr of the above
expression gives:
- 49 -
• • • • • (3-19)
which is the same as equation (3-16).
b) é total based on tube-side flow, with Ct in denominator,
The expressions on the R.H.S. of (3-20) cannot be formed into
effectiveness , however, by substituting 2Csh for Ct as before gives:
.E = c{ ll-t• -l:n) + ln., -n.)} + 4U ... - b-,) s Z Cst..l b.., - tt-t>) '2-(st.l k:s.\ -ho)
~-rf ~( (ln - tb) -t( f:t,_ tto) -+
l eSt.. ( t~\ ~h-o)
which is the same as equation (3-16) •
• • • • • (3-21)
The expressions (3-14) to (3-21) will apply in the limits
or
Possibility 5:
cs/ ct
but csh ..(ct
- 50 -
a) é total based on shell-side flow,
c~ l ts, - ts.). :t~~) Et= -----:-----::-
Ct- l t~,- tt-c) • • • • • (3-22)
=
• • • • • (3-23)
Since Ct is not emin for individual sections of the heat exchanger,
therefore, the expression (3-23) does .not represent the sum of the individual
effectivenesses of the exchangers.
The required result can he obtained by multiplying (3-23) thrpugh-
out in the denominator and numerator by Csh'
c~ c~t. . .C t~,- t ~~ ) = Cst._ ')(. (~ ( k-s,- ho)
Cst. -c~
~ c~Ct~- bol.) t Cg_ >< Ct ltS\- tt-6)
••••• P-24)
-51. -
b) é total based on tube-aide flow,
[_ . = Ct (tt~- b-o) t Lt- ( ts, - tto )
• • • • • (3-25)
Ct t(h-~ -tt'l-)+(b-,- ho)} Ct( tt-l.- f:~) = + -----
4( t~,- tM~) Ct ltcra,- t~:o) • • • • • (3-26)
This again is not applicable, but by a similar process as for
(3-23), multiplying the denominator and numerator throughout by C8h, gives
us,
• • • • • (3-27)
which is the same as equation (3-24).
The expressions (3-24) and (3-27) being applicable in the limit
csh - = 0.5 - 1. Ct
- 52 -
It will be interesting to note that in case of
or
= 1, the expressions (3-24) and (3-27)
reduce to
l.. [ ~ l:s.\ - b, .. , \ Es = 2 C..2.st.._
4 é., fs, - b.-o j which is the same as for
Possibility 4, and confirms equations (3-14), (3-16), (3-19) and (3-21).
Confirming (3-24) and (3-27) for the other limit,
Csh ct = 1,
for which the same e~ressions reduce to
which is the same as for
Po_ssibility 2, and confirma equations (3-6) and (3-8).
In all the foregoing expressions .in this article, for the total
tsl - ttl effectiveness of the split-flow exchanger, a common factor . t is
tsl - to encountered which must be expressed in terms of dimensionless parameters
emin r.--- and NTUmax, in order that a complete expression in terms of these îllax
dimensionless quantities may pe obtained.
:DA 2G e... from (2-23a)
from (2-27b)
- 53 -
"' Ct )c;+ct ( c~)
z e. \ ' - E.'Z. <4. ==-----__;_--------~~ . . . . . (3-28)
Dividing (3-28) by ey in the numerator and denominator,
=
- 54 -
J In equation (3-29) the value ofF will be (2-30b), while in equation
J (3-30) the value ofF will ' be (2-30c).
tsl - ttl Before substituting these values of in the equations of
tsl - tto overall effectivenesses, the various possibilities represented by (3-2) to
(3-27) may be summarized~
Ct for --- = 0 - 1,
csh
csh for --- = 0 - .5
ct
csh for --- = 0. 5 - 1
ct
• • • • • (3-31)
••••• (3-32)
• • • • • (3-33)
The equation (3-33) shall not be used since in .order to calculate
r:8
, equation (3-32) is sufficient.
A very vital point to be noted in (3-31) and (3-32) is that while
calculating the total weighted effectivenesses, in case of (3-31) the values
of [ 2t and f:1 are to be calculated with half of the total area of the split
flow exchanger, i.e. with Ah in NTUmax. On the other band, in case of (3-32),
the values of[ 2sh andE 1 are to be calculated with total area of the split
flow exchanger, i.e. using A in NTUmax•
Bearing in mind the above, the equations (3-34) and (3-35) for the
total effectiveness can be written as given in the next pages.
.v
ct _ Catin.. for Csh - Cmax and where NTUmax =
UAh ~in (also in E2t and El)
tKTU-( ~~ -}+ (~n ( 1 _ E1t ~~) ~ ~ é, 2 e_ -KTIJ~I+ ('d tl-= élt +· ---
-tm~_Jt+(~J 1 + e..
-IITU-Ji~ \-e. +-
}tr---+ (-t:-T 1-
~ \- c~
t+e. . ,_ -} (~\~ -ffrU~t+ (~::) h -~- x~
1 ~ ~
-N~I+ (ê.,_j 1 ~ +-(~ 1+ e....
-~;::::=::=:==---- + --
-NTYI+ (t:l } +(~ '- Q_
(3-34)
VI VI
csh emin for-= r. -ct -max
c é:2Sl.. + cs:c:-~
·~·
and where NTUmax "" ~n (also in[ZSh andE 1)
+ rrry,_( 1-)1+ (~J)
E., e_
-liTt)._}+ (~S -HT~JI+(~ t-e..
~+Q.. +-----
\+ (é)~
( \- é2.~
c~ -c~
J rc~r -tmJ~ 1+\~ ~ -1 ~
1+ e. c.,.;..\ ... -J tC....\ -IITU-d 1+ (c....J 1+ IJ;.:}
1-!L
~l+(~f ~+1 c~
'+L +--
. -N~t+(§f )+(?J 1-~
•••••••••• (3-3S)
VI o-
- 57 -
The values of effectiveness have been found out for various
values of NTUmax and Cmin for (l-31), (2-32), (3-34) and (3-35). These Cm a x
values were obtained with the help of Fortran programmes, run in McGill
University's Computing Centre, and the effectiveness curves are plotted.
(Pages 78 and 79). Cmin From the curves of effectiveness we note that for ---- = 0 1 ~x
the effectiveness of the reverse flow exchanger is the same as that of
the split-flow exchanger at any given value of NTUmax· For all other
Cm in values of r.---1 the effectiveness is greater for reverse flow exchanger
"1JlaX
than for split-flow exchanger and this becomes quite marked beyond NTUmax
of about 2, .
In the split-flow exchanger itself, the values of effectiveness
Cs for-= 0
Ct Ct
and Cs = 0 at all values of NTUmax· The same is are the same
Cs true for -ct
Ct = 1 and C = 1 at all values of NTUmax•
s However, for any
other value of capacity rate ratios the effectiveness of a split-flow
exchanger is higher for
Ct value of C and NTUmax •
s values of ~ greater
Cs any value of - and NTUmax than for the same ct This difference becomes quite pronounced for
than about 3.
From the above it can be concluded that from the point of view
of thermal effectiveness, the lawer capacity fiuid should be placed on
the shell side of the split-flow exchanger.
- · 58 -
IV - Petermin~tion of Correction Faètor Ft•
Having found the final thermal effectiveness êt and Es, it is now
quite simple to calculate the Ft from basic rate equations.
q = UA Ft L.M.T.D. (counter-flow)
= UA F t (!:s,- tr~)-( h4- b:o)
l ts,- tt~ tsLt- tto
where t ~ ;- ts~ 9.4 = 7.
...... (4-1)
(4-2)
• • • • • ( 4-3)
• • • • • (4-4)
:. from the above four equations,. it can be derived that
1-RP UA ln ï:'P
= Ct Ft (1-R) • • • • • ( 4-5)
• • • • • ( 4-6)
and
..... (4-7)
- 59 ..
bence substituting (4-2), (4-6) and (4-7) in (4-5),
• • • • • ( 4~8)
For the case when Ct = Cmin and Cs = Cma:x
• • • • • ( 4-9)
For the case when Cs = ~in and Ct = ~ax
((~ ) NrU --\ 'h\.M(. c~
emin when -.-- = 1, the ab.ove two equations become
. cmax
f '-t
••••• (4-11)
- 60 -
Values of Ft in both the cases have been computed and curves
f F h b d ith NTU b i .and Cmin as the third o t ave een rawn w max as a sc ssa Cmax
parameter. (Page 80)
- 61 -
V - Experimental Equipment
G~ne;al
A split-flow exchanger of ·single shell and 4 tubeet .2-pass .design
was built and set up in the research laboratory. of the Mechanical De~art~ent • . ·
It was decided to have air in the shell-side a.a well aa the -tube-
side of the exchanger. The maximum ~ir supply c.apacity of the ·:laboratory'a
compressors is about 1,000 lbs/hr. With this amount of air supply1 sufficiently
high values of the overall beat transfer coefficient U ·could be obtained with
a .. shell ·aiZe of 1 3/4 inch diameter .( 16 BWG),and four 1/2 inch , \20 :st<m)tubes
inside the shell as shown in the cross-section Figure - 9. -To provide
suffi.cient beat transfer area (to obtain a high value .of NTUmax ~ ~n) .the,
exchanger was made 12 feet long.
Exchanger Construction
The exchanger was made from braaa tubes and brasa parts to . faci~it•te
machining and soldering.
Baffles:
40tfo eut segmental baffles from .1/1.6'' thic:k s\leet were rQ&de. The
baffles were spaced 6 inches apart, held securely .by means of baffle spacers
which consist of 3/16" thick thr.ough-b.olts serewed into the tube sbeet and
6" long 3/16" diameter 21 :SWG spacer tubes •
. Air System
The main air was .supplied .at about 100 lbs/S.q." gauge pressure.
Ai r Filter:
In order t~ pnrify the air from atmospheric dirt, rust carried from
old pipe-lines and moisture, an air filter 1 foot diameter and 2 1/2 feet long
packed with ~otton-waste was f itted in the main line.
- 62 -
SPLIT-FLOW BEAT EXCHANGER
1 3/4" diameter, 16 BWG Shell
- 1/2" diameter, 20 BWG Tubes
Equally spaced on 7/8" Pitch Circ1e
Cross-section of She11 and Tube Arrangement
FIGURE - 9
- 63 -
Air Heater:
The air heater was made out of 4 inch diameter, 5 feet long
standard pipe. Ten tubular heating elements of 750 watts each, cQnnected
in parallel with a terminal voltage of 120 volts D.C. were fitted in the
pipe. This total capacity of 7.5 kw was sufficient to raise the temperature
of 600 lbs/br of air through about 200 °F.
Hot air was run in the tube-side of the exchanger.
Air metering:
The:outlets of both the shell aides were connected to a single pipe
and led to exhaust through 3" diameter 6 feet long standard pipe •. At the
centre of this exhaust pipe sharp-edged orifice with D and D/2 tappings was
made. Orifice plates of sizes 3/4", 1", 1 1/4",_ 1 1/2" and 1 3/4" diameter.
were made to insert the suitable aize to give a reasonable difference of
water column in the manometers depending upon the flow rate.
For tube-side, a similar arrangement as above was made for the
flow measurement.
The orifiees were made according to British Standard Code
B.S.l042: 1943, and they were not calib~ated.
A line diagram of the flow system is shawn in Figure - 10.
Temperature Measurement
The temperatures were measured by 19 gauge iron-constantan thermo
couples connected to a precision potentiometer. The thermocouples were
calibrated in ice-cold water as well as boiling water and a curve was pre
pared for each thermocouple to effect adjustment:.
- 64 -
A line diagram of the apparatus indicating the thermocouple
stations is shown in Figure - 11.
Insulation
The exchanger was insulated with lagging material of about
0 0.028 BTU/hr. sq.ft F/ft. thermal conductivity.
Ge~eral views of the beat exchanger set up are shown in
Figures 12 and 13.
SPLIT-FLOW HEAT EXCHANGER
AIR FLOW ARRANGEMENT
7.5 kW Tube-Side Air Heater
/Tube Inlet +
Air Main
Tube-Side Gate Valve
+ 120 V D.C. Supply
Shell Out let ____,
-=-===--=----===-== ===-::: =-=-==--=--= =-~ -=-=-..: = =-:.-::::---~= .=.=.:.:;o Orifice Shell Inlets ~~ ~4 _ l/20 20 BWG Tubes Sliding Tube E
3/40, 16 BWG Shell 1
ff) 1
1
/
\Tube Out let
Tube Flow Control Valves
FIGURE - 10
Air Filtér
Main Air Gate Valve
t
"' Ut
SPLIT-FLOW HE.AT EXCHANGER
16
8
.5
4 5
~~~------------------~~~----------------~~J~l2
LOCATION AND NUMBERING .OF THERMOCOUPLES
FIGURE - 11
0\ 0\
- 67 -
SPLIT-FLOW HEAT EXCHANGER
Precision potentiometer for thermocouples
Carbon resistance control for the air heater
Voltmeter and
Manometer Board for orifices and pitot tubes.
Shell-side flow control valves
Tube-side flow control valves
CONTROL PANEL
FIGURE - 12
- 68 -
SPLIT-FLOW HEAT EXCHANGER
air heater
in let
TUBE-SIDE INLET AND OUTLET
FIGURE - 13
an er
carbon resistance control for heater
out let
" .
3.
- 69 -
VI - Precautions
The following main precautions were observed in running the tests:
1. The main air supply was from three compressors, two of 250 lbs/hr.
eacp and one of 500 lbs/hr. capacity. All the three compressors
when switched on are controlled bY aut.omatic switches adjusted for
100 lbs/sq.in. gauge pressure of the receiver tank.. Unless full
air is utilised, due to off-and-on of the compressors, the air
supply is fluctuating. Therefore, a bleed valve was fitted on the
line and adjusted in such a way that full supply i .s drawn and all
the three compressor.s remain operating.
2. In order to have equal flow in bath aides of the shell, _one valv.e
in each outlet was fitted. Pitot tubes were fitted 6" before the
valves. Pitot probes were made to determine the flow in each half
and thereby establish equal flow in both halves of the shell by
regulating it with the valves.
However, since the traverse was over a short length of 1.049" (the
inside diameter of l" standard pipe) and the traverse being manually
operated, an accuracy of not more than about 5% could be achieved in
equally dividing the shell-side flow.
The temperature values required in the exchanger were for the bulk,
therefore, the sensing elements of the thermocouples were made large
enough to c_over t'he full stream. At sorne stations (3-4, 5-6,
17-18, 19·20) two thermocouples were fitted to give a more average
value of the temperature.
- 70 -
At the centre of the tube lengths 1where there could b~ maximum
variation in temperature across the cross-section, a special
c.onstruction of the form shown below was made to create turbu-
lence in the fluid in order to measure the average temperature
of the fluid.
small dive plate
diameter tub.e
FIGURE - 14
4. Each set of observations wàs made after steady-state conditions
had reached about two hours.
- 71 -
VII - Test Resulta
Following are the sample results of a number of tests performed.
c min/C ; max Effectiveness Effectiveness
No. Ct/Cs cs/ct NTUmax Experimental Theoretical deviation
1 0.42 2.51 o. 742 0.745 0.40i
2 0.36 2.·97 o. 753 0.795 5.30
3 0.354 2.76 0.756 0.79 4.20
4 0.428 2.65 o. 730 0.745 2.00 '
5 0.560 i 2.24 0.670 0.690 2.90
6 0.91 1.63 0.580 0.555 4.5
7 o. 708 1.76 0.648 0.615 5.37
' 8 0.432 2.25 ' 0.768 0.730 5.20
9 0.288 3.37 o. 790 0.825 4.25
10 0.345 3.04 0.772 0.80 3.50
11 0.386 2.84 o. 750 0.775 3.225
12 0.460 2.55 o. 717 0.735 2.45
13 0.462 2.70 0.694 0.740 6.22
14 ' 0.326 2.34 0.817 o. 78 4.75
15 0.825 1.98 0 .• 600 0.602 0.33
- 72 -
Samp1e Ca1culation
Following is a typical data of one of the runs with air flow:
Shell side flow W8 = 314 lbs/hr.
C8
= W8 x Cp = 314 x .24 = 75.5 BTU/hr °F
Tube aide flow Wt = 381 lbs/hr.
- 0 ct = wt x cP = 381 x .24 = 91.5 BTU/hr F.
Temperatures at various cardinal points are as indicated below:
241.5
184.5
175.8·----- 163.4 164
He at Balance: FIGURE - l5
Shell side: q = Cs (ts4 - tsl)
= 75.5 (184.5 ; 164. - 74.3)
= 7550 BTU/hr.
Tube si de: q = ct (tt3 - tto)
= 91.5 (241.5 - 163 .4)
= 7140 BTU/hr
... 73 -
Percentage difference in heat balance = 5.3.
Heat transfer area of the exchanger = 6.04 sq.ft.
Overall heat transfer coefficient U:
The overa11 heat transfer coefficient in both ba1ves of the
exchanger is assumed to be the same. U for the 1eft band side can be
calculated aa fo11ows:
NTU : max
Log-mean temperature difference = 57 .6
p tube side temperature difference 36.8 = total temperature difference = 124.7 = •296
R = shell side temperature difference 89.7 2 46 tube side temperature difference = 36.8 = •
Ft (1-2 exchanger) = .79
u = csh (ts2 - tsl> .LMTD x Ft x Ah
314/2 x.2.4 (164 ~ 74.3) 57.6 x .79 x 3.02
= 24.5 BTU/hr ft 2 °F.
= 314 x .24 = 75.5 BTU/hr °F
Ct = 381 x .24 = 91.5 BTU/hr °F
Cmin = 75.5 BTU/hr °F
emin c · max
75.5 = 9ï':"5 = .825
UA =--emin
24.5 x 6.04 = 75.5
= 1.985
Effectiveness:
actual heat transferred = maximum possible heat transferred
= 75.5 184.7 ; 73.3 - 23.5}
75.5 (116.3 - 23.5)
= 0.60
- 74 -
VIII - Discussions and Concl~sions
From the curves of effectivenes.s Figure - 19 and Figure - 20,
the following observations are made:
1. The effectiveness of a split-flow exchanger is practically the
same a.s that for a reverse-flow exchanger for values of NTUmax
below about 2. However, for values of NTU a greater than about mx
2, the effectiveness of a split-flow exchanger is less than that
of. a reverse flow exchanger.
2. In case when one fluid is isothermal, whether it is in the tube-
side or in the shell-side the effectiveness of a split-flow
exchanger is the same a.s that of a reverse-flow exchanger or a
pure counter-flow exchanger.
3. In a split-flow exchanger, for NTU more than about 2.5, the max
effectiveness is greater when the smaller capacity fluid is i .n
the .shell and greater eapacity in the tubè-side. This is a very
useful rèsult since most_of the split-flow exchangers are liquid-
gas excpangers, and the shell-side always carries the gas . for
pressure drop considerations and fortunately this arrangement
givea .a better ~hermal effeetiveneas as well.
4. Thermodynamically, a split-flow exchanger gives highest effective-
ness in the region of NTU between about 3 and 5 and decreases max
beyond that. The reason for this is .as follows:
- 75 -
Consider the temperature distribution as shown in Figure - 16.
FIGURE - 16
The ultimate temperature difference is tsl - tto• The maximum heat
exchange takes place in sections of greatest temperature difference. Consider
ing the tube side, the highest heat exchange is in Section I and lowest in
Section IV.
If the beat transfer area of the beat exchanger be increased beyond
a certain value, thereby increasing NTUmax' most of the heat energy having been
exchanged in Sections I, II and III, the Section IV might actually be cooled
instead of being heated particularly if there is a temperature cross as shown
in Figure- 17. This means that the additional beat transfer area
FIGURE - 17 tto
in Section IV bas actually reduced the total amount of beat transferred thereby
reducing the overall effectiveness.
5. The performance of a split-flow exchanger is not altered by reversing
the flow directions or by causing the shell to carry the cold stream
and the tubes to carry the hot stream. This is due to the fact that
the effectiveness is a function of the exchanger configuration, ~ax'
- 76 -
Cmin Cmax' and reversing the fluid direction& .. and causing either of the
shell or tube-side to carry hot or cold stream does not change any
of the variables and therefore cannat alter the effectiveness.
From the experimental resulta, the the.oretic.al curves of effectiveness
+ ti. have been checked and found correct with an average of about - 3.510 error. This
confirmation, however, is true only for values of NTUmax below about 3. With
air-flow, NTUmax values of greater than about 3.5 could not be obtained due t.o
small overall heat tranafer coefficiento It is therefore recommended that .with
the same exchanser water-water combinat ion should b.e tried. For thi.s certain
alterations will be necessary e.g. iron-constantan thermocouples will have to
be changed to avoid corrosion or the thermocoup-le tips will have to be protected
from contact with water, a water heater will be required etc.
The Correction Factor Ft
The graph of correction facto.r Ft 1 Figure - 21 1 shows. that for the
same values of Rand P, there is very little difference, "ab.out 1/2~ or less,
between a split-flow exchanger and a reverse-flow exchanger for values of Ft
greater than about 0.8. This, of course, is due to the fact that there is
practically no difference in effectiveness between the two exchangers for NTUmax
values of le.ss than about 2.
Further Developments
1. In this analysis, a two tube pass exchanger hàs been deal~.witb. · -It is
possible to extend the w.ork to cover 4 or more tube passes.
2. A divided-flow exchang~r with two or more tube passes can also be
ana~ysed in a similar way as the split-flow exchanger.
FIGURE - 18
1.
2.
3.
4.
5.
6.
Nagle, W.M.
Underwood, A.J.V.
Bowman, R.A.
ftusselt, W.
Nusselt, W.
Smith, D.M.
- 77 -
References
Ind. Eng. Chem. Vol. 25 1 p. 604 (1933)
J. Inst. Petroleum Tech. Vol. 20, p. 145 (1934)
Ind. Eng. Chem. Vol. 28, p. 541 (1936)
Zeitschrift des Vereines deutscher lnginieur
Vol. 55, p. 2021 (1911)
Technische Mechanik und Thermodynamik Vol •. l,
p. 417 (1930)
Engineering Vol. 1381 p. 479 and p. 606 (1934)
7. Bowman, R.A.; A.C .• Mueller and W .M. Nagle
8.
9.
10.
Kreith, F.
T·en Broeck, H.
Trans. Am. Soc. Mech. Engrs. Vol. 62, p. 283 (1940)
Principles of Heat Transfer, p. 453
International Text Book Company, Scranton
Ind. Eng. Chem. Vol. 30, p. 1041 (1938)
(1960)
Kays, W.M. and A.L. London
Compact Heat Exchangers, p. 7
McGraw-Hill Book Company, N.Y. (1958)
11. Kern, D.Q. and C:~L. Carpenter
Chem. Eng. Progress Vol 47, p. 211 (1951)
12. Schindler; D.L. and H.T. Bates
13. Wright, c.e.
Chem. Eng. Progress Symposium Series Vol. 561
p. 203 (1960)
Para11e1-counter flow shell-tube beat exchangers.
Pe~formance in terms of effectiveness, capacity
rate ratio and number of beat transf er units 1
unpublished Stanford University Mechanical
Engineering report (1954)
0.7
0.6 'fil. l.tJ • • u
~ -::!0.5 u u ... ... ~
0
0.3
o·.2
0
0.1
- 78 -
Thermal Effectiveness
SPLIT-FLOW EXCHANGER
NTUmax
Cs -Ct
Ct --- c.
No. of Transfer Unite, NTUmax = AU/Cmin
UA a Catin
Cm in = c_-;
Cmin c Snax
t/<.0.6
UJ • • ~ 1>
:0.5 u
" ... ... ...
0.4
1
1'
.,
! 1.1
1;· ,,
1 1 ;
1 l' . 1'
- 79 -
,,
•l·
' 1 •1.11 ' ~ : :11:1;: ~::. ;~
,:
1 1 1 ,
Il 1 1
1 ,. .
Ill 11- IHI Il Ill
Thermal Effectiveneel
SPLIT-FLOW IXCHANGIR v1. REVERSI-FLOW EXCHANGER
l C1 C...in
.,,,,_,,,. ......... --------- :: -:::: c. • Ciaax
c. ct Cain ! Reveree-Flow Excbanaer • • • • • • • • • • Ct • Ce • Cux
1.
l-1 o. 0 .u (J
~
""' Q) (J
s:: Q)
~o. ~ ~ ..-1 ~
Q)
l-1 :l ., Qj
~o. ~ Q)
E-4
"' .u
""' o.
o.
"
~ =""-- ,.. "" ~ l'.:'
9 rnrn~~~*ffimR~~~~ffmmm ...
~,rnrn~~~~ttm~8~~#m~ttm ~œrnmm~#m~mH~lliW~#m~mm
1 ~ ~
7 ' ' ' l
_l j_,
: 1 1 1 ' j_ · - -~ i ' 1 - 0
1 ' ' 1 • ' ' .... . _.~ 1 ..... .o
t-+-+-1-~1-U-'-"' H o a fd.-_ ~ , • ~ 1.0 _ ~
6 1
5·.·--~··· 0 0.1 0.2 0.4 0.6 0.3 0.5
p
0.7 0.8
~T~! IXl'D CORRECTION . FACTOR, Ft
J SPLIT-FLOW EXCHANGER
~ -t2
~1 Tl - T2 t2 - tl
' T2 = ts2 + ts3 R = p = ta3 t2 - tl Tl - tl
t•2 2
FIGURE - 21
0.9 1.0
j
(X) 0
_u~(c .. )2 q'+tc;_ 1 + e..
~=-_UA~Ç2l' c:-)2
(~ ] l+l~
j_~
. . ,· :~··?~-~-· . . . . -.:· .. ~
., .'
. ;
•
l- c~ ( Cst. !
) \...
------ ---------·- -·--· --------- - --
y -Y C (e_ - a_) ~
f+ .. .
' ( .. + ~tJ 1
l
J ·-------
_UAL Ç(S!:)i. t4 _ t cs,t.. P;- '4 c~
1-t-e... ___ - --
-UA~ 1-+ (Cs..t )-:;_ J l(<t)z. ~ c.. 1 + \C.~
1- Q.
-fuJI+ fC~t...)L Gt.. c~.._ \ct ( +1
e_ t 1 +- ------- + -==
.J!&_ 1 CSl-.)7. r. j '_Q_ c"' '+ (c,:- J 1 +~?:
f. = --- ------------------- ------;!_ - uA._I + (~j cs.(
c.~l c~ - -1 Ct-
\+e.. --- - --
-~Fl~) }~cii:l ·-~
•
!
y
~ - -· ... ----~---- --- .. ------------- .. ----
)
:. --. ~ ~:· - ~ ' .· ~,. ~ ~~ ,,. .,.